NormOps (dense64 0.27 API)

java.lang.Object
- org.ejml.ops.NormOps

```
public class NormOps
extends java.lang.Object
```
Norms are a measure of the size of a vector or a matrix. One typical application is in error analysis.

Vector norms have the following properties:
1. ||x|| > 0 if x ≠ 0 and ||0|| = 0
2. ||αx|| = |α| ||x||
3. ||x+y|| ≤ ||x|| + ||y||
Matrix norms have the following properties:
1. ||A|| > 0 if A ≠ 0 where A ∈ ℜ ^{m × n}
2. || α A || = |α| ||A|| where A ∈ ℜ ^{m × n}
3. ||A+B|| ≤ ||A|| + ||B|| where A and B are ∈ ℜ ^{m × n}
4. ||AB|| ≤ ||A|| ||B|| where A and B are ∈ ℜ ^{m × m}
Note that the last item in the list only applies to square matrices.

Matrix norms can be induced from vector norms as is shown below:

||A||_M = max_x≠0||Ax||_v/||x||_v

where ||.||_M is the induced matrix norm for the vector norm ||.||_v.

By default implementations that try to mitigate overflow/underflow are used. If the word fast is found before a function's name that means it does not mitigate those issues, but runs a bit faster.

Constructor Summary

Constructors
Constructor and Description

NormOps()

Constructors
Constructor and Description
`NormOps()`

Method Summary

All Methods Static Methods Concrete Methods
Modifier and Type	Method and Description
`static double`	`conditionP(org.ejml.data.DenseMatrix64F A, double p)` The condition number of a matrix is used to measure the sensitivity of the linear system Ax=b.
`static double`	`conditionP2(org.ejml.data.DenseMatrix64F A)` The condition p = 2 number of a matrix is used to measure the sensitivity of the linear system Ax=b.
`static double`	`elementP(org.ejml.data.RowD1Matrix64F A, double p)` Element wise p-norm: norm = {∑_i=1:m ∑_j=1:n { \|a_ij\|^p}}^1/p
`static double`	`fastElementP(org.ejml.data.D1Matrix64F A, double p)` Same as `elementP(org.ejml.data.RowD1Matrix64F, double)` but runs faster by not mitigating overflow/underflow related problems.
`static double`	`fastNormF(org.ejml.data.D1Matrix64F a)` This implementation of the Frobenius norm is a straight forward implementation and can be susceptible for overflow/underflow issues.
`static double`	`fastNormP(org.ejml.data.DenseMatrix64F A, double p)` An unsafe but faster version of `normP(org.ejml.data.DenseMatrix64F, double)` that calls routines which are faster but more prone to overflow/underflow problems.
`static double`	`fastNormP2(org.ejml.data.DenseMatrix64F A)` Computes the p=2 norm.
`static double`	`inducedP1(org.ejml.data.DenseMatrix64F A)` Computes the induced p = 1 matrix norm. \|\|A\|\|₁= max(j=1 to n; sum(i=1 to m; \|a_ij\|))
`static double`	`inducedP2(org.ejml.data.DenseMatrix64F A)` Computes the induced p = 2 matrix norm, which is the largest singular value.
`static double`	`inducedPInf(org.ejml.data.DenseMatrix64F A)` Induced matrix p = infinity norm. \|\|A\|\|_∞ = max(i=1 to m; sum(j=1 to n; \|a_ij\|))
`static void`	`normalizeF(org.ejml.data.DenseMatrix64F A)` Normalizes the matrix such that the Frobenius norm is equal to one.
`static double`	`normF(org.ejml.data.D1Matrix64F a)` Computes the Frobenius matrix norm: normF = Sqrt{ ∑_i=1:m ∑_j=1:n { a_ij²} }
`static double`	`normP(org.ejml.data.DenseMatrix64F A, double p)` Computes either the vector p-norm or the induced matrix p-norm depending on A being a vector or a matrix respectively.
`static double`	`normP1(org.ejml.data.DenseMatrix64F A)` Computes the p=1 norm.
`static double`	`normP2(org.ejml.data.DenseMatrix64F A)` Computes the p=2 norm.
`static double`	`normPInf(org.ejml.data.DenseMatrix64F A)` Computes the p=∞ norm.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - NormOps
```
public NormOps()
```
- Method Detail
  - normalizeF
```
public static void normalizeF(org.ejml.data.DenseMatrix64F A)
```
    Normalizes the matrix such that the Frobenius norm is equal to one.
    
    Parameters:
    
    A - The matrix that is to be normalized.
  - conditionP
```
public static double conditionP(org.ejml.data.DenseMatrix64F A,
                                double p)
```
    The condition number of a matrix is used to measure the sensitivity of the linear system Ax=b. A value near one indicates that it is a well conditioned matrix.
    
    κ_p = ||A||_p||A^-1||_p
    
    If the matrix is not square then the condition of either A^TA or AA^T is computed.
    
    Parameters:
    
    A - The matrix.
    
    p - p-norm
    
    Returns:
    
    The condition number.
  - conditionP2
```
public static double conditionP2(org.ejml.data.DenseMatrix64F A)
```
    The condition p = 2 number of a matrix is used to measure the sensitivity of the linear system Ax=b. A value near one indicates that it is a well conditioned matrix.
    
    κ₂ = ||A||₂||A^-1||₂
    
    This is also known as the spectral condition number.
    
    Parameters:
    
    A - The matrix.
    
    Returns:
    
    The condition number.
  - fastNormF
```
public static double fastNormF(org.ejml.data.D1Matrix64F a)
```
    This implementation of the Frobenius norm is a straight forward implementation and can be susceptible for overflow/underflow issues. A more resilient implementation is normF(org.ejml.data.D1Matrix64F).
    
    Parameters:
    
    a - The matrix whose norm is computed. Not modified.
  - normF
```
public static double normF(org.ejml.data.D1Matrix64F a)
```
    Computes the Frobenius matrix norm:
    
    normF = Sqrt{ ∑_i=1:m ∑_j=1:n { a_ij²} }
    
    This is equivalent to the element wise p=2 norm. See fastNormF(org.ejml.data.D1Matrix64F) for another implementation that is faster, but more prone to underflow/overflow errors.
    
    Parameters:
    
    a - The matrix whose norm is computed. Not modified.
    
    Returns:
    
    The norm's value.
  - elementP
```
public static double elementP(org.ejml.data.RowD1Matrix64F A,
                              double p)
```
    Element wise p-norm:
    
    norm = {∑_i=1:m ∑_j=1:n { |a_ij|^p}}^1/p
    
    This is not the same as the induced p-norm used on matrices, but is the same as the vector p-norm.
    
    Parameters:
    
    A - Matrix. Not modified.
    
    p - p value.
    
    Returns:
    
    The norm's value.
  - fastElementP
```
public static double fastElementP(org.ejml.data.D1Matrix64F A,
                                  double p)
```
    Same as elementP(org.ejml.data.RowD1Matrix64F, double) but runs faster by not mitigating overflow/underflow related problems.
    
    Parameters:
    
    A - Matrix. Not modified.
    
    p - p value.
    
    Returns:
    
    The norm's value.
  - normP
```
public static double normP(org.ejml.data.DenseMatrix64F A,
                           double p)
```
    Computes either the vector p-norm or the induced matrix p-norm depending on A being a vector or a matrix respectively.
    
    Parameters:
    
    A - Vector or matrix whose norm is to be computed.
    
    p - The p value of the p-norm.
    
    Returns:
    
    The computed norm.
  - fastNormP
```
public static double fastNormP(org.ejml.data.DenseMatrix64F A,
                               double p)
```
    An unsafe but faster version of normP(org.ejml.data.DenseMatrix64F, double) that calls routines which are faster but more prone to overflow/underflow problems.
    
    Parameters:
    
    A - Vector or matrix whose norm is to be computed.
    
    p - The p value of the p-norm.
    
    Returns:
    
    The computed norm.
  - normP1
```
public static double normP1(org.ejml.data.DenseMatrix64F A)
```
    Computes the p=1 norm. If A is a matrix then the induced norm is computed.
    
    Parameters:
    
    A - Matrix or vector.
    
    Returns:
    
    The norm.
  - normP2
```
public static double normP2(org.ejml.data.DenseMatrix64F A)
```
    Computes the p=2 norm. If A is a matrix then the induced norm is computed.
    
    Parameters:
    
    A - Matrix or vector.
    
    Returns:
    
    The norm.
  - fastNormP2
```
public static double fastNormP2(org.ejml.data.DenseMatrix64F A)
```
    Computes the p=2 norm. If A is a matrix then the induced norm is computed. This implementation is faster, but more prone to buffer overflow or underflow problems.
    
    Parameters:
    
    A - Matrix or vector.
    
    Returns:
    
    The norm.
  - normPInf
```
public static double normPInf(org.ejml.data.DenseMatrix64F A)
```
    Computes the p=∞ norm. If A is a matrix then the induced norm is computed.
    
    Parameters:
    
    A - Matrix or vector.
    
    Returns:
    
    The norm.
  - inducedP1
```
public static double inducedP1(org.ejml.data.DenseMatrix64F A)
```
    Computes the induced p = 1 matrix norm.
    
    ||A||₁= max(j=1 to n; sum(i=1 to m; |a_ij|))
    
    Parameters:
    
    A - Matrix. Not modified.
    
    Returns:
    
    The norm.
  - inducedP2
```
public static double inducedP2(org.ejml.data.DenseMatrix64F A)
```
    Computes the induced p = 2 matrix norm, which is the largest singular value.
    
    Parameters:
    
    A - Matrix. Not modified.
    
    Returns:
    
    The norm.
  - inducedPInf
```
public static double inducedPInf(org.ejml.data.DenseMatrix64F A)
```
    Induced matrix p = infinity norm.
    
    ||A||_∞ = max(i=1 to m; sum(j=1 to n; |a_ij|))
    
    Parameters:
    
    A - A matrix.
    
    Returns:
    
    the norm.

Class NormOps

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

NormOps

Method Detail

normalizeF

conditionP

conditionP2

fastNormF

normF

elementP

fastElementP

normP

fastNormP

normP1

normP2

fastNormP2

normPInf

inducedP1

inducedP2

inducedPInf